This invention deals with a particular kind of heterogeneous system, which can be described as a porous body consisting of a continuous solid matrix with embedded pores that can be filled with either gas or liquid. According to S. Lowell et al, the spatial distribution between the solid matrix and pores can be characterized in terms of a porosity and pore size. According to J. Frenkel and M. A. Biot, the mechanical properties of such systems with respect to any applied oscillating stress depend primarily on the visco-elastic properties of the matrix. Lyklema notes that when such porous bodies are saturated with liquid, additional properties are then related to any surface charge on these pores, which in turn is commonly characterized by a zeta potential. Although methods exist for characterizing these mechanical and electrical properties, they all have limitations and call out for improvement. Here we suggest a method for determination of pore sizes using electroacoustic measurement.
Propagation of ultrasound through a wetted porous material generates electric response that is known as seismo-electric effect. There is also reverse effect generated with oscillating (AC) electric field when it is applied to a wetted porous material. It generates ultrasound wave. This effect is called electro-seismic effect. Both of the effects belong to the family of electroacoustic phenomena according to ISO standard 13099 Part 1.
Both of the electroacoustic effects occur in porous materials due to existence of electric charges on surfaces of the wetted probe and screening diffuse layers located in the liquid next to the surfaces. This structure called electric double layers [5]. The double layers are characterized with certain thickness called Debye length and there is a potential drop across them—zeta potential (ζ). There are several papers dedicated to theoretical and experimental studies of these effects [7-23].
It turns out that these electroacoustic phenomena can be used for characterizing porous materials, in particular, pore size, Zhu et al. 2008 [15]. Applicants elaborate on this application for the seismo-electric current in the instant application. The present invention is valid for electro-seismic effect as well.
Measurement of the seismoelectric current presents no problem with present commercially available instruments, as it was shown in papers by Dukhin et al. in 2010 [11, 22, 23].
Still, interpretation presents certain challenges. There is no general theory that would be valid for any combination of parameters. Instead, there are several theories, each of them limited to a certain range of parameters. This is a result of the multiple effects that might take place inside of the pores when ultrasound propagates through them. Therefore, interpretation becomes a main element of characterization procedure. Different pore size ranges require different interpretation procedures, which might be subjects for different patents.
Applicants show herein that interpretation could make characterization procedure independent of the calibration with material having known pore size. Such absolute methods are very rare and very valuable due to simplicity and high reliability.
There are three distinctively different ranges of pore sizes. Each of them requires different interpretation procedure for calculating pore size from the measured signal.
Range of Small Pores with Overlapped Double Layers.
This range covers pore size up to 200 nanometers. Consequently it includes porous materials with micropores, pore size <2 nm; mesopores, pore size is between 2 and 50 nm and small macropores, with pore size >50 nm and <300 nm.
Thickness of the electric double layers inside of the probes can be comparable with size of the pores in this range. As a result, double layers located on the opposite sides of the pore overlap. This overlap can be enforced by reducing ionic strength of the wetted liquid. Debye length of the double layers formed in the distilled water does not exceed 50 nm. This limits size of the pores with double layers overlap.
There is U.S. Pat. No. 8,281,662 issued Oct. 9, 2012 to Dukhin, Goetz and Thommes claiming application of the overlapped double layers mode for characterizing pore size. The method is not absolute, it requires calibration with material having known pore size.
Range of Intermediate Pore Size, Smoluchwski Range [6].
There is no pore size dependence of the measured electroacoustic signal. This is the range where Smoluchwski type electroacoustic theories are valid. Such theories are summarized in the ISO standard 13099 Part 1, 2012. This range is the most suited for determining zeta potential of pores, instead of pore size.
Values of pore sizes that belong to this range depend on the frequency of ultrasound. For the frequency of 1 MHz, this range encompasses pores with sizes roughly 200 nm up to 1 micron. Higher frequency would shift top limit of the pore size range to lower values.
Range of Large Pores with Deviation from Steady Hydrodynamic Poiselle Profile.
Space distribution of the hydrodynamic flow through a pore generated by a constant gradient of pressure is described by parabolic Poiselle profile [3, 4]. This distribution remains valid for oscillating gradient of pressure if frequency of oscillation is low enough. Then Poiselle profile becomes established at any point of time due to slow changing pressure gradient.
This steady state distribution becomes distorted when frequency of oscillation (ultrasound frequency f) becomes high enough. According to Biot [3, 4] one may assume that the Poiselle flow breaks down for a given frequency when the pore size d becomes larger than a certain critical value dcr:
                              d          >                      d            cr                          =                              πν                          4              ⁢                                                          ⁢              f                                                          (        1        )            
where ν is kinematic viscosity.
If pore size exceeds this critical value for a given frequency (see Eq. 3), then the hydrodynamic flow oscillates too fast, steady state of the Poiselle profile cannot be reached, and then the seismo-electric current becomes dependent on the pore size. For water and the frequency 3 MHz, this is expected to occur for the pore diameter above approximately 0.5 micron.
It turns out that at the larger pore sizes where the hydrodynamic relaxation effect dominates the seismo-electric current decreases with increasing pore size. This is opposite to the effect of the overlapping double layers. There is experiment published in the paper by Dukhin et al in 2013 that confirms this dependence. FIG. 1 demonstrates results of this experiment. This effect can be separated from the effect of the overlapped double layers by selecting wetting/conducting liquid with sufficiently high ionic strength.
This experiment was performed on a series of monolithic resorcinol based organic aerogels, which serve as pre-cursor for carbon aerogels, see Reichenauer in 2005. These materials exhibit essentially identical porosity (ca. 70%) but vary in pore diameter from ca. 1 um up to 8 um. The pore diameters were determined by mercury porosimetry.
This experiment confirmed theoretically predicted role of the hydrodynamic relaxation in the seismoelectric effect.
There is at least one known analytical theory that describes effect of hydrodynamic relaxation on electroacoustic in pores. Repper and Morgan in 2002 derived following equation for electroacoustic effect in porous material:
                                          Δ            ⁢                                                  ⁢                          P              ⁡                              (                ω                )                                                          Δ            ⁢                                                  ⁢                          V              ⁡                              (                ω                )                                                    =                                            2              ⁢              ɛζκ                        a                    ⁢                                                                      J                  1                                ⁡                                  (                                      κ                    ⁢                                                                                  ⁢                    a                                    )                                                                              J                  0                                ⁡                                  (                                      κ                    ⁢                                                                                  ⁢                    a                                    )                                                                                    -                1                            +                                                2                                      κ                    ⁢                                                                                  ⁢                    a                                                  ⁢                                                                            J                      1                                        ⁡                                          (                                              κ                        ⁢                                                                                                  ⁢                        a                                            )                                                                                                  J                      0                                        ⁡                                          (                                              κ                        ⁢                                                                                                  ⁢                        a                                            )                                                                                                                              (        2        )            
where ΔP and ΔV are gradients of the seismic pressure and voltage at the angular frequency ω=2πf, ∈ is dielectric constant, α is pore radius, J is Bessel function, and
                    κ        =                                                            -                i                            ⁢                                                          ⁢              ω                        ν                                              (        3        )            
where i is imaginary unit.
According to this theory, seismoelectric and electroseismic signals at the frequency where hydrodynamic relaxation becomes significant should be expressed in terms of complex numbers. Critical frequency that determines this range can be calculated from the condition:κα=1  (4)
which leads to the following expression for the critical angular frequency for a given pore radius:
                              ω          cr                =                              2            ⁢            π            ⁢                                                  ⁢                          f              cr                                =                      ν                          a              2                                                          (        5        )            
where fcr is critical frequency in MHz.
For pore diameter equals to 1 micron this critical frequency is roughly 1 MHz.
The critical frequency determines approximately a middle point of the range where hydrodynamic relaxation is important. We can assume that complete frequency range encompasses frequency within order of magnitude higher and lower than this critical frequency.
There was suggestion of using this theory for calculating pore size from the measured electroacoustic signal published by Zhu et al in 2008. However, there is no explanation how to conduct this calculation. This theory contains at least one unknown parameter—zeta potential (ζ) of pores surfaces. This parameter depends on chemistry of the pore surfaces, composition of the wetting liquid.
Idea suggested by Zhu et al in 2008 cannot be accomplished unless method would be suggested for dealing with this unknown parameter. That is why applicants consider that such additional method, allows elimination of this unknown parameter from the characterization procedure.
Measured electroacoustic signal is complex number that has creation magnitude and phase. Equation 2 reflects this fact. Second multiplier with Bessel functions on the right hand side of the equation is a complex function of the parameter κα. For simplicity Eq. 2 is written as follows:
                                          Δ            ⁢                                                  ⁢                          V              ⁡                              (                ω                )                                                          Δ            ⁢                                                  ⁢                          P              ⁡                              (                ω                )                                                    =                                            2              ⁢              ɛζκ                        a                    ⁢                                    H              *                        ⁡                          (                              κ                ⁢                                                                  ⁢                a                            )                                                          (        6        )            
where complex function H* equals:
                                          H            *                    ⁡                      (                          κ              ⁢                                                          ⁢              a                        )                          =                                                                              J                  1                                ⁡                                  (                                      κ                    ⁢                                                                                  ⁢                    a                                    )                                                                              J                  0                                ⁡                                  (                                      κ                    ⁢                                                                                  ⁢                    a                                    )                                                                                    -                1                            +                                                2                                      κ                    ⁢                                                                                  ⁢                    a                                                  ⁢                                                                            J                      1                                        ⁡                                          (                                              κ                        ⁢                                                                                                  ⁢                        a                                            )                                                                                                  J                      0                                        ⁡                                          (                                              κ                        ⁢                                                                                                  ⁢                        a                                            )                                                                                                    =                                    Re              ⁢                                                          ⁢                              H                *                                      +                          i              ⁢                                                          ⁢              Im              ⁢                                                          ⁢                              H                *                                                                        (        7        )            
Measurement yields information for the magnitude and phase of the electroacoustic signal.
Theoretical expression for the magnitude is following:
                              Magnitude          ⁡                      (                                          Δ                ⁢                                                                  ⁢                                  V                  ⁡                                      (                    ω                    )                                                                              Δ                ⁢                                                                  ⁢                                  P                  ⁡                                      (                    ω                    )                                                                        )                          =                                            2              ⁢              ɛζκ                        a                    ⁢                                                                      (                                      Re                    ⁢                                                                                  ⁢                                          H                      *                                                        )                                2                            +                                                (                                      Im                    ⁢                                                                                  ⁢                                          H                      *                                                        )                                2                                                                        (        8        )            
It is seen that magnitude depends on the unknown parameter—zeta potential. Therefore, calculation of pore size from the electroacoustic magnitude would require calibration procedure with the same material having known pore size. Such calibration reference material can be prepared only using independent characterization method, mercury porosimetry as an example. This leads to the conclusion that calculation of pore size from the measured magnitude of the electroacoustic signal is not an absolute method.
Theoretical expression for the phase of the electroacoustic signal is quite different:
                              Phase          ⁡                      (                                          Δ                ⁢                                                                  ⁢                                  V                  ⁡                                      (                    ω                    )                                                                              Δ                ⁢                                                                  ⁢                                  P                  ⁡                                      (                    ω                    )                                                                        )                          =                  arctan          ⁢                                    Im              ⁢                                                          ⁢                              H                *                                                    Re              ⁢                                                          ⁢                              H                *                                                                        (        9        )            
It is seen that the phase is independent of unknown zeta potential. Actually it would be independent on all parameters that can be presented as real numbers. They simply cancel out. This is a tremendous advantage of using phase instead of magnitude of electroacoustic signal for calculating pore size. This method would become absolute, requiring no calibration with material having known size.
This method of calculating pore size from the phase of electroacoustic signal has important advantages over alternative methods—calculation from the magnitude.
This method differs from our previous patent on this subject, the U.S. Pat. No. 8,281,662 B2 by Dukhin, Goetz and Thommes due to difference in the interpretation procedure. Our previous patent was designed for overlapped double layers and calculation of the pore size from the magnitude of the electroacoustic signal. The new proposed patent suggests calculation of the pore size from the phase of the electroacoustic signal, which is pore size dependent due to hydrodynamic relaxation in larger pores at high frequency.
This method can be applied for determination of pore size distribution as well. It would be possible if measurements conducted at multiple frequencies. Frequency dependence of electroacoustic phase contains information on the pore size distribution.